reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem Th36:
  for G1,G2 being Group
  for H1 being Subgroup of G1
  for K being Subgroup of H1
  for H2 being Subgroup of G2
  for f being Homomorphism of G1,G2
  for g being Homomorphism of H1,H2
  st (for k being Element of G1 st k in K holds f.k=g.k)
  holds Image(f|K) = Image(g|K)
proof
  let G1,G2 be Group;
  let H1 be Subgroup of G1;
  let K be Subgroup of H1;
  let H2 be Subgroup of G2;
  let f be Homomorphism of G1,G2;
  let g be Homomorphism of H1,H2;
  assume A1: for k being Element of G1 st k in K holds f.k=g.k;
  A2: Image(f|K) is strict Subgroup of G2 &
      Image(g|K) is strict Subgroup of G2 by GROUP_2:56;
  for y being object
  holds y in the carrier of Image(f|K) iff y in the carrier of Image(g|K)
  proof
    let y be object;
    thus y in the carrier of Image(f|K) implies y in the carrier of Image(g|K)
    proof
      assume y in the carrier of Image(f|K);
      then consider h being Element of K such that
      B1: (f|K).h = y
      by STRUCT_0:def 5,GROUP_6:45;
      B2: h is Element of G1 & h is Element of H1 & h in K by GROUP_2:42;
      f.h = g.h by A1,B2
         .= (g|K).h by B2, Th1;
      then (g|K).h = (f|K).h by B2,Th1
                  .= y by B1;
      hence y in the carrier of Image(g|K) by STRUCT_0:def 5,GROUP_6:45;
    end;
    thus y in the carrier of Image(g|K) implies y in the carrier of Image(f|K)
    proof
      assume y in the carrier of Image(g|K);
      then consider h being Element of K such that
      C1: (g|K).h = y
      by STRUCT_0:def 5,GROUP_6:45;
      C2: h is Element of H1 & h is Element of G1 & h in K by GROUP_2:42;
      g.h = f.h by A1,C2
         .= (f|K).h by C2,Th1;
      then (f|K).h = (g|K).h by C2,Th1
                  .= y by C1;
      hence y in the carrier of Image(f|K) by STRUCT_0:def 5,GROUP_6:45;
    end;
  end;
  hence Image(f|K) = Image(g|K) by A2,GROUP_2:59,TARSKI:2;
end;
